1,926 research outputs found
Record process on the Continuum Random Tree
By considering a continuous pruning procedure on Aldous's Brownian tree, we
construct a random variable which is distributed, conditionally given
the tree, according to the probability law introduced by Janson as the limit
distribution of the number of cuts needed to isolate the root in a critical
Galton-Watson tree. We also prove that this random variable can be obtained as
the a.s. limit of the number of cuts needed to cut down the subtree of the
continuum tree spanned by leaves
Asymptotics for the small fragments of the fragmentation at nodes
We consider the fragmentation at nodes of the L\'{e}vy continuous random tree
introduced in a previous paper. In this framework we compute the asymptotic for
the number of small fragments at time . This limit is increasing in
and discontinuous. In the -stable case the fragmentation is
self-similar with index , with and the results are
close to those Bertoin obtained for general self-similar fragmentations but
with an additional assumtion which is not fulfilled here
A construction of a -coalescent via the pruning of Binary Trees
Considering a random binary tree with labelled leaves, we use a pruning
procedure on this tree in order to construct a -coalescent
process. We also use the continuous analogue of this construction, i.e. a
pruning procedure on Aldous's continuum random tree, to construct a continuous
state space process that has the same structure as the -coalescent
process up to some time change. These two constructions unable us to obtain
results on the coalescent process such as the asymptotics on the number of
coalescent events or the law of the blocks involved in the last coalescent
event
Assume-Admissible Synthesis
In this paper, we introduce a novel rule for synthesis of reactive systems,
applicable to systems made of n components which have each their own
objectives. It is based on the notion of admissible strategies. We compare our
novel rule with previous rules defined in the literature, and we show that
contrary to the previous proposals, our rule defines sets of solutions which
are rectangular. This property leads to solutions which are robust and
resilient. We provide algorithms with optimal complexity and also an
abstraction framework.Comment: 31 page
Local limits of galton-watson trees conditioned on the number of protected nodes
We consider a marking procedure of the vertices of a tree where each vertex
is marked independently from the others with a probability that depends only on
its out-degree. We prove that a critical Galton-Watson tree conditioned on
having a large number of marked vertices converges in distribution to the
associated size-biased tree. We then apply this result to give the limit in
distribution of a critical Galton-Watson tree conditioned on having a large
number of protected nodes
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